Remark 2.2 (Monge-Amperè bracket as a weak sequential limit) In this remark, we show that the generalization of the Monge-Amperè bracket for certain distributional fields is the weak sequential limit of the Monge-Amperè bracket of smooth fields. We consider two fields \(B_1\)  and \(B_2\) , both in \(\mathcal {B}(\Omega )\) , such that \(\nabla \nabla B_2\)  concentrates on the interface whereas \(\nabla \nabla B_1\)  is a piecewise continuous function which does not concentrate on the interface; these regularity assumptions are stronger than those considered in REF  where the Hessian of both the fields was allowed to concentrate on the interfacial curve.
In more formal terms, let \(B_1\in \mathcal {B}(\Omega )\)  with bulk density \(b_1\)  be such that \(\nabla \nabla B_1 \in \mathcal {B}(\Omega ,\operatorname{Sym})\)  is square integrable, i.e., \(\nabla \nabla b_1 \in L^2(\Omega ,\operatorname{Sym})\) , and \(B_2 \in \mathcal {B}(\Omega )\)  with bulk density \(b_2\)  be such that \(\nabla B_2 \in \mathcal {B}(\Omega ,\operatorname{Sym})\)  is square integrable, i.e., \(\nabla b_1 \in L^2(\Omega ,\mathbb {R}^2)\) .
For sequences \({b_1}^n \in C^\infty (\Omega )\)  and \({b_2}^n \in C^\infty (\Omega )\) , such that \(\nabla \nabla {b_1}^n \rightharpoonup \nabla \nabla {b_1}\)  weakly in \(L^2(\Omega ,\operatorname{Sym})\)  and \(\nabla {b_2}^n \rightharpoonup \nabla {b_2}\)  weakly in \(L^2(\Omega ,\mathbb {R}^2)\) , \([{b_1}^n,{b_2}^n] \rightarrow [B_1,B_2]\)  in the sense of distributions.
Indeed, \(\operatorname{curl}(\nabla {b_1}^n \otimes \nabla {b_2}^n)=(\langle -\partial _{12} {b_1}^n e_1 +\partial _{11} {b_1}^n e_2, \nabla {b_2}^n \rangle ) e_1+(\langle -\partial _{22} {b_1}^n e_1 +\partial _{12} {b_1}^n e_2, \nabla {b_2}^n \rangle ) e_2.\)  We note that \(\operatorname{curl}(\nabla {b_2}^n)=0\)  and \(\operatorname{div}(-\partial _{12} {b_1}^n e_1 +\partial _{11} {b_1}^n e_2)=\operatorname{div}(-\partial _{22} {b_1}^n e_1 +\partial _{12} {b_1}^n e_2)=0\) . The Div-Curl lemma [1]} then implies that \(\operatorname{curl}(\nabla {b_1}^n \otimes \nabla {b_2}^n) \rightarrow \operatorname{Curl}(\nabla {b_1} \otimes \nabla {b_2})\)  in the sense of distributions. Using \([{b_1}^n,{b_2}^n]=- \operatorname{curl}\operatorname{curl}(\nabla {b_1}^n \otimes \nabla {b_2}^n),\)  we have \([{b_1}^n,{b_2}^n] \rightarrow -\operatorname{Curl}\operatorname{Curl}(\nabla {b_1} \otimes \nabla {b_2})\)  in the sense of distributions.

